The purpose of this paper is to survey some of the recent finiteness results on rational and integral points on algebraic varieties defined over global fields of positive characteristics.
In the classical case of number fields, the basic results are the Mordell-Weil theorem, the Thue-Siegel-Roth theorem and Siegel's theorem on integral points on curves (see [L1]). Recently, there have been some major developments, first Faltings proved the Mordell conjecture, then Vojta gave another proof and this latter method was extended by Faltings [F3,4] to prove two outstanding conjectures of Lang. There are some historical comments in [F2] and [L2] is a general survey of the subject. In the number field case there are also very general quantitative conjectures made by Vojta [Vj]. These seem very deep and beyond the reach of current techniques. Of course, one should consider also the case of function fields of characteristic zero and there has been considerable progress there also. We refer to [L2] and [B2] for more details. In [B2] there is also a description of a tentative transposition of the function field methods to the number field case.
The situation in positive characteristic is similar to that of number fields.
Results similar to those of Faltings have been proved recently, by different
methods. We shall describe these results in detail below.
One would hope to have conjectures similar to Vojta's in the function field
case. However, the direct transposition of Vojta's conjectures to the
case of positive characteristics is false. It also seems that any
simple modification of the conjectures has counterexamples. We have been
unable to formulate a plausible conjecture, even in the case of 
.
We will present below several examples that illustrate pathological behaviour
in positive characteristic and perhaps a more perspicacious reader will find
a general pattern in which these examples will fit.
We shall not attempt a full description of the history of the subject but let
us point out a few highlights before proceeding to a description of recent
results. The analogue of the Mordell-Weil theorem was proved by Lang and
Neron (see [L1]) guided by Severi's remark that it was related to his Theorem
of the Base.
Diophantine approximation was
considered by Mahler and Osgood initially but the results are still fragmentary.
(See section 4). In particular, Mahler showed that
Liouville's inequality cannot be improved in
general and so Siegel's argument to deal with integral points on curves over
number fields (which were successfully transferred to the case of
function fields of characteristic zero by Lang) cannot be applied in
characteristic p.
It is unclear who first proved that elliptic curves in characteristic p with
non-constant j-invariant have finitely many integral points. It seems to
be well-known that this follows
formally from the Mordell conjecture, (see e.g. [B1], ch 7,
thm. 3.1) which was proved
first by Samuel ([Sa]) but it is unclear who first noticed
this formal consequence.
The result was proved by Mason ([Ma], proof of thm. 14, pg. 114)
and another proof given by the author [V4]. As mentioned above,
the Mordell conjecture was proved by Samuel ([Sa1,2]), extending Grauert's
proof for function fields of characteristic zero to characteristic p.
Szpiro then gave an effective proof ([Sz]) as a consequence of his work on the
Shafarevich conjecture. Let us also mention that recently Pheidas ([Ph])
proved that the problem of deciding if a polynomial in several variables and
coefficients in 
has a zero with coordinates in 
is
unsolvable. As usual, this has not abated the search for positive results
in this area.
We do not, strictly speaking, follow Weil's advice in the above quotation. In fact, most of the recent success in the positive characteristic case is due to exploiting the special circumstances of this case.
A fundamental notion in this study is that of isotriviality. Roughly speaking, one is interested in deciding whether a variety defined over a function field can be defined over its constant field. We devote an appendix to this notion, studying it from different points of view.
Throughout this paper, K will denote a global field of positive characteristic p. In other words, K is a function field in one variable over a finite field of characteristic p.
This is an updated version of a paper published in the Proceedings of the Cortona conference in Arithmetic Geometry, F. Catanese, ed., Symposia Mathematica.